(d-i) (3%) Find p (0,1)R(p,pU) p (0,1)R(p,pB) , and p (0,1)R(p,pT) for a fixed n

題組內容

3. (30%) Let X₁, X₂, ..., Xₙ be a random sample from Bernoulli(p). Define R(p, δ(X)) = Eₚ[(δ(X) - p)²] (the Mean-Square Error).
(d) Consider three estimators

$$\hat{p}_U = \text{UMVUE found in (b)},$$

$\hat{p}_B$ = the posterior mean with constant risk found in (c-iii),

and

$$\hat{p}T = \begin{cases} \hat{p}U \text{ with probability } \frac{n}{n+1}, \\ 1/2 \text{ with probability } \frac{1}{n+1}. \end{cases}$$

一、下面這一段來自犯罪學者 George Vold 的《理論犯罪學》，*Theoretical Criminology*，請做個理論說明與評論。(25 分) The Anomie Tradition The term anomie is derived from an ancient Greek word meaning “lawlessness.” In criminology it is important because of its use by Emile Durkheim and Robert K. Merton in their discussions on the structure of society and the effects that structure can have on individuals within society. These writers used “anomie” to designate a state of “normlessness” in society which could have the effect of generating deviant behavior in individuals. Like Lombroso, who had written his major work approximately 20 years earlier, Durkheim was reacting against the classical assumptions that man was free in a contractual society. But whereas Lombroso focused on the determinants of human behavior within the individual, Durkheim focused on society and its organization and development. Durkheim’s original formulation of anomie was later revised by Merton, who applied it directly to American society. In this form, it gained widespread acceptance, and subsequently provided the ideological underpinning for Lyndon Johnson’s War on Poverty.

題組內容

(d-i) \( (3\%) \) Find \( \sup{p\in (0,1)}R(p,\hat{p}U) \) \( \sup{p\in (0,1)}R(p,\hat{p}B) \) , and \( \sup{p\in (0,1)}R(p,\hat{p}T) \) for a fixed \( n \)

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